where \(S_{\omega }x=x-2(x\cdot \omega )\omega \)  is a reflection.
This of course implies that \(W:X\rightarrow X\)  is bounded for any function space \(X\)  on \({\mathbb {R}}^3\)  with a norm which is invariant under translations and reflections.
The proof of this representation formula in [1]} is not entirely straightforward. On the one hand, the algebra to which the Wiener theorem is applied
is somewhat delicate and requires casting the finite order Born series terms in Yajima's work [2]} (which involve only finitely many potentials and free resolvents) in
some iterative algebraic framework. In other words, one needs to find the correct algebra \(\mathcal {A}\)  and composition law \(\circledast \)  as well as operator \(T\)  to write the third Born term, say, in the form \(T\circledast T\circledast T\)  in \(\mathcal {A}\) . Furthermore, the classical scattering theory
based on weighted \(L^2\)  spaces does not suffice and it is necessary to invoke the the author's work with Ionescu [3]}, which revisits the classical Agmon-Kato-Kuroda theorem
in the context of Fourier restriction and the Stein-Tomas theorem, as well as the
Keel-Tao endpoint [4]}. This in turn relies on the Carleman theorems and absence of imbedded eigenvalues obtained in [5]}.
It is not known whether a structure theorem holds under a scaling invariant assumption on \(V\) ,
see however [6]} for such a result, albeit involving small scaling-invariant potentials.
